Improved Upper Bounds for the Largest Eigenvalue of Unicyclic Graphs
Received:June 04, 2007  Revised:July 07, 2008
Key Words: unicyclic graph   adjacency matrix   largest eigenvalue.  
Fund Project:the National Natural Science Foundation of China (No.10861009).
Author NameAffiliation
HU Sheng Biao Department of Mathematics, Qinghai Nationalities College, Qinghai 810007, China 
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      Let $G(V,E)$ be a unicyclic graph, $C_{m}$ be a cycle of length $m$ and $C_{m}\subset G$, and $u_{i}\in V(C_{m})$. The $G-E(C_{m})$ are $m$ trees, denoted by $T_{i}$, $i=1,2,\ldots,m$. For $i=1,2,\ldots,m$, let $e_{u_{i}}$ be the excentricity of $u_{i}$ in $T_{i}$ and $$e_{c}=\max\{e_{u_{i}}: i=1,2,\ldots,m\}.$$ Let $k=e_{c}$ 1. For $j=1,2,\ldots,k-1$, let $$\delta_{ij}=\max\{d_{v}:{\rm dist}(v,u_{i})=j, v\in T_{i}\},$$ $$\delta_{j}=\max\{\delta_{ij}:i=1,2,\ldots,m\},$$ $$\delta_{0}=\max\{d_{u_{i}}:u_{i}\in V(C_{m})\}.$$ Then $$\lambda_{1}(G)\leq \max\{\max\limits_{2\leq j\leq k-2}(\sqrt{\delta_{j-1}-1} \sqrt{\delta_{j}-1}),2 \sqrt{\delta_{0}-2},\sqrt{\delta_{0}-2} \sqrt{\delta_{1}-1}\}.$$ If $G\cong C_{n}$, then the equality holds, where $\lambda_{1}(G)$ is the largest eigenvalue of the adjacency matrix of $G$.
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